Projections under seminorms and generalized Moore Penrose inverses

Abstract

The definition of a projector under a seminorm is given. Such a projector is not unique. Operators projecting into a given linear subspace under a seminorm form an affine linear subalgebra of the linear associative algebra of square matrices. The authors have introduced elsewhere the concept of a minimum seminorm semileast squares inverse of a complex matrix. It is shown here that the same concept could also be defined in terms of projectors under seminorms. This extends a similar definition for the Moore Penrose inverse given in terms of orthogonal projectors under the usual Euclidean norms. Various properties of a projector under a seminorm and also of a minimum seminorm semileast squares inverse are obtained including representations giving general solutions for both.